Nber Working Paper Series Capital Utilization And

NBER WORKING PAPER SERIES

CAPITAL UTILIZATION AND CAPITAL ACCUMULATION: THEORY AND EVIDENCE

Matthew D. Shapiro

Working Paper No. 1900

NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts Avenue Cambridge, MA 02138 April 1986

The research reported here is part of the NBER'S research program in Economic Fluctuations. Any opinions expressed are those of the author and not those of the National Bureau of Economic Research. NBER Working Paper #1900 April 1986

Capital Utilization and Capital Accumulation: Theory and Evidence

ABSTRACT

A firm may acquire additional caoital ini-ut by nurchasing new cai- tal or by increasing the utilization of its current capital. The margin between capital accumulation and capital utilization is studied in a model of dynamic factor demand where the firm chooses capital, labor, and their rates of utilization. A direct measure of capital utilization--the work week of capital--is incorporated into the theory and estimates. The estimates imply that capital stock is costly to adjust while the work week of capital is essentially costless to adjust. The estimated response of the capital stock to changes in its price and in the required rate of return is more rapid than found in other estimates.

flatthew D. Shapiro Cowles Foundation for Research in Economics Yale University Box 2125 Yale Station New Haven, CT 06520 1. Introduction

In this paper, I examine the link between capital utilization and

the dynamic demand for capital. A firm can vary the intensity of use of

its capital by lengthening its work week as well as by changing the work

week and amount of labor. When a firm decides--for whatever reason--that

it needs more capital services than it already has, it has two options.

It may either acquire additional physical capital or use its current capi-

tal more intensively. As long as the firm is not at a corner, it has these

two choices regardless of whether the quantity of labor is increased correspondingly. That is, the choice of amount of capital services can be made separately from the choice of labor input and the capital-labor ratio.

The Federal Reserve Board's and the Bureau of Economics Analysis's published measures of capacity utilization are not necessarily economically meaningful measures of capital utilization. Each measures actual output relative to potential rather than at what rate of intensity capital is being used.

In this paper, I discuss an alternative measure of capital utilization, the work week of capital. By assuming that capital is idle if no labor is present, the work week of capital can be deduced from data on shift work of labor.

I discuss construction of such a series, present some results about its statistical properties, and integrate it into a model of dynamic factor demand.

I estimate the model of dynamic factor demand and study how capital accumulation and capital utilization respond to changes in the cost of capital.

The aim in this paper is to analyze and quantify how the firm's abil- ity to vary capital utilization impinges on its demand for capital. Why firms use their capital so little is a persistent puzzle. The main purpose of this paper is not to resolve this puzzle, but rather to analyze how the

1 margin between extending shift work and adding new capital affects the de-

cision to buy new capital. Nonetheless, my estimates are consistent with

the view that the productivity of shift work is low.2

Uncertainty about the future plays a central role in the firm's in-

vestment decision. For example, whether the firm exoects a shock to be

permanent or temporary will determine whether it responds mainly by increas-

ing its capital stock or by increasing the utilization of the current stock.

I use the research strategy advocated by Hansen and Singleton (1982) of

estimating the stochastic first-order conditions of the firm's decision

problem. This methodology allows expected future outcomes to drive current

decisions without requiring the econometricians to characterize fully their

distribution.

To simulate the estimated model, I make further assumptions about

the environment facing the firm. I can then solve using the algorithm of

Blanchard and Kahn (1980) for the estimated response of capital demand and capital utilization to changes in factor prices. The current response of demand and utilization depends on current and future values of factor prices and the required rates of return. These changes can arise because of changes in tax policy. Hence, the estimates can be used to study changes in tax policy based on estimated structural parameters of the representative firmts obj ective function.

In Section 2, I discuss the work week of capital as a measure of capacity utilization. In Section 3, I discuss the theory of interrelated factor demand with variable factor utilization. I present estimates of the model in the fourth section. In the fifth section, I consider some extensions concerning the overidentifying restrictions of the model and the aggregation problem. I explore in Section 6 the dynamic response of capital 3

to changes in costs induced, for exanirle, by changes in tax policy.

The work week of capital overshoots the steady state when prices and rates

of return change. The estimated response of capital stock to changes in

prices and required rates of return is substantial. This result provides

an important challenge to the standard view that prices and required rates

of return are empirically unimportant in models of the demand for capital.

I summarize the results in the last section.

2. Neasuring Capital Utilization

There is no officially published time series directly measuring capi-

tal utilization in U.S. manufacturing. "Capacity utilization" as published

by the FRE, BEA, or in the Wharton index is a measure of actual output rela-

tive to potential, not a measure of capital utilization. This point is

stressed not to criticize the FRB, BEA, or Wharton indexes per se but to

characterize their economic content. In particular, they are not good meas-

ures of the intensity of use of the canital stock as maintained by some re-

searchers [Nadiri and Rosen (1969) and Tatoin (1980), for example]. I discuss

in the Appendix how these measures of capacity utilization are essentially

detrended output. The correlation of detrended value-added in manufacturing

and the FRB index is 0.88.

The measure of capital utilization I propose to use is the average work week of capital. The series is analogous to the well-knownaverage work week of labor, Ht , measuredin man-hours. Capital is working only if some labor is present, that is, if the plant is open. Published information on shift work is therefore used to construct the implied work week of capital.

The Bureau of the Census in 1929 collected and again since 1973 4

collects information directly on the work week of capital [see Foss (1981) for details]. These data are too infrequent and not collected over a long enough continuous time span to be useful in this study.

The measure of the work week of capital I use here is based on the nuer of workers on late (second and third) shifts in manufacturing. Data on shift work are available beginning in the early 1950s from the Area Wage

Surveyconductedby the Bureau of Labor Statistics. The survey gives the percentage of workers on the second and third shifts in the various SMSAs.

and Let L1 , L2 , L3 denote employment on the first, second, and third shifts. Consider a series

S = - + - + (1) [H(L1L2) 80(L2L_) 120L31/L1

The series S measures the work week of capital. Equation (1) is equiv- alent to Taubman and Gottschalk's (1971) equation (2.3) up to a normalization. Suppose that only one shift is worked so L2 and L3 are both zero.

In this case S ,thework week of capital, equals Fl , thework week of labor. Increasing the utilization of capital is achieved exactly byin- creasingthe utilization of labor. The index is constructed so that overlapping shifts are not double-counted. In particular, itassumesthat overtimework on one shift overlaps with that on another shift if more than one shift is worked. This insures that S does not overstate the length of the work week of capital. In particular, it helps to ensure that S is not spuriously correlated with the work week of labor, which is an alternative indicator of capital utilization.3

Now suppose that two equally-sized shifts are worked so that L1 and L2 are equal and L3 is zero. Then the work week of capital is eighty hours. (No information is available on the length of late shifts so aver- 5 agehours are used.) If three equal shifts are worked, then the work week of capital is 120. It is presumed that if more than one shift works, they are andL3 can be deduced from of equal size. Therefore, L1 , L2, aggregate production workers, L ,andfromthe shares of shift work in theArea Wage Survey. Taubman and Gottschalk (1971) construct a quarterly index S for 1952 through 1968 based on Area Wage Survey data. The Survey is conducted fordifferent SNSAs at different points intime. They use this information to construct a quarterly series from the disaggregate data. I extend the series through 1982 basedon published and unpublished summary statistics for the post 1968 period.4I then use a modification of the Chow-Lin (1969) procedure for best linear unbiased distribution. The instrument for the distribution is quarterly average hours. I estimate the relationship between quarterly hours and S on the 1952 to 1968 period. From these estimates, I construct fitted values for quarterly S after 1968 by redistri- buting the annual residual equally over each quarter of the year. The entire series is given in the Appendix.

Use of an interpolated series creates an errors-in-variables problem.

In Section 5.2, I discuss estimates that are robust to measurement error in S

There are shortcomings to use of this measure of the work week of capital as a measure of capital utilization. First, because of data limit- ations, it does not account for work on weekends. In 1976, the average plant in the U.S. manufacturing industry operated 5.3 days per week [Foss

(1981, p. 9)] so at least on average weekend work is negligible. Second, the measure is for capital in use so it does not account for temporary plant closings. Permanent plant closings should, in principle, be captured in 6

the measured capital stock. Finally, the work week of capital is a better measure of capital utilization in assembly industries such as automobiles than in process industries such as blast furnaces and petroleum refining.

In process industries, it may be extremely costly to completely shut down a plant. Hence, utilization maybevaried by changing the quantity of materials processed.

Before presenting the theoretical model of the interrelation between the stock demand and utilization of factors, it is useful to consider the correlation of the work week of capital with other macroeconomic variables.

For the following discussion, I detrend the data with linear and quadratic trends.5 The correlation of the work week of capital with output is 0.66, so it is strongly pro-cyclical. The correlation of the work week of production workers with output is 0.70; the utilization rates of capital and labor are about equally correlated with output. The two utilizations have correlation 0.77.

The correlation of the work week of capital with labor productivity

(output per man-hour) is 0.13. Longer hours for capital means a higher effective capital-labor ratio, and consequently higher labor productivity.

The correlation of the capital stock with the sane measure of labor productivity is 0.08,sohigh frequency changes in productivity are more substantially correlated with the work week than the stock of capital. Foss (1981) emphasizes the importance of changes in capital utilization for the trend determination of productivity. Tatom (1980) notes that variation in capital utilization might explain pro-cyclical labor productivity. The correlations presented here are consistent with capital utilization being part of the source of pro-cyclic productivity. Tatom uses FRB capacity utilization to proxy for capital utilization, which, for reasons I discuss above, does not directly measure capital utilization. 7

3. Factor Utilization and the Theory of Dynamic Factor Demand

The technology I study allows the firm to choose independently its

capital and labor stocks and their rates of utilization. The firm takes

factor and product prices as given. The level of output is determinedby

the firm's choice of inputs and their utilizations rather than by anexog-

enous constraint. Thus, the choice of the rate of utilization determines

output, and not vice versa. The scope to vary utilization gives the firm

flexibility in responding to shock to demand and cost. In particular, it

may respond to temporary shocks principally by adjusting utilization and

to permanent shocks principally by adjusting the stock of a factor.

The effect of varying utilization on maximized profits can be di-

vided into four parts. First, under typical concavity assumptions, increas-

ing the utilization of one factor should decrease the marginal product of

that factor. Second, increased factor utilizationmay impose added vari-

able cost on the firm. For example, to increase labor utilization a firm

must pay the wage rate plus a possible premium for overtime. To add a shift

the firm may pay a shift premium. Third, as I demonstrate, changing util-

ization or quantities employed by different factors implies different

adjustment costs. These different adjustment costs have different impli-

cations for profits. Specifically, the adjustment cost of capital is sub-

stantial and significant, that of employment is smaller yet significant, and that of hours of employees and of capital is nil. Fourth, therate of utilization may affect user cost of capital by changing the rate ofdepre- ciation Isee Taubman and Wilkinson (1970)]. Haking the rate ofdeprecia- tion endogenous has the unfortunateconsequence of making the level of the capital stock unobservable. I do not examine the implications of utilization for user cost in the context of this paper's model. 8

Nadiriand Rosen (1969) study capacity utilization in a model where

finns are not explicitly forward-looking. Theystudydynamically interre-

lated factor demands for labor and capital where the level of utilization

of both capital and labor are choice variables of the firm. They use a

Cobb-Douglas production function where capital, capacity utilization, em-

ployees, and hours of employees all have different shares. Nadiri and Rosen

use capacity utilization to measure capital utilization. Measured capacity

utilization is essentially output divided by potential. Consequently, capac-

ity utilization as measured cannot be an independent choice of the finn.

Itis therefore misleading to estimate an independent "demand" for it.6

The literature on interrelated factor demand under rational expectations largely overlooks the issue of capacity utilization. That is, it is typically assumed that the current stock of factors is always utilized fully. An exception is theoretical work by Abel (1979), whose model is close to the one in this paper. Sargent (1978) estimates a dynamic demand function for labor. He does in a sense address the issue of utilization by estimating separate demands for straight-time and overtime employment.

He finds that straight-time employment has a cost of adjustment far in excess

(e.g., 40 times) that for overtime employment. His conclusion regarding the different cost of adjustment of overtime and straight-time work is similar to a result in this paper: that the number of employees is costly to adjust but that the number of hours they work is not. He does not, however, study demand for capital or its utilization.

In other studies of dynamic factor demand under rationalexpectations, factors are utilized fully. Meese (1980) estimates the interrelated demands for capital and man-hours. Kennan (1979) estimates the demand for labor 9

as a function of current and expected future output. Pindyck and Rotemberg

(1983) separate white-collar from blue-collar employment but again assume that each is utilized at a constant rate.

I now consider an explicit model of the firm that yields demands for both capital and labor and their rates of utilization.

3.1. Technology

This section extends the theory of interrelated factor demand to account explicitly for the firm's separate choice of factor utilization. /

Inaddition to choosing its stock of capital, Kt, itsstock of production workers, Lt , andtheir hours, Ht ,andits stock of non-production workers, N ,itcanalsochoose the number of shifts it operates. The number of shifts is proportional to the work week of capital, St .That is, the number of shifts is essentially the ratio of to Ht If the firmchoosesto extend the work week of capital, S ,without changing labor input, it would correspondingly reduce the effective capital-to-labor ratio. Thus, one would expect, but not require, that the firm add labor when it adds shifts. If the work week of capital is extended

by lengthening the work week of production workers, , thenno corres- I-It ponding increase in labor is required to keep fixed the ratio of workers to machines. Recall from Section 2 that to chose the work week of capital the firm in fact choses the amount of work on late shifts. To emphasize this connection I refer to S as the work week of capital, but the reader is free to think of an increase in S as the decision to move workers from the early to late shifts. In any case, S is a choice variable of the firm independent of the total production worker employment, Lt

To make consideration of the technology concrete, consider the pro- 10

ductionfunction to he used in the empirical work. I use a Cobb-Douglas

function augmented bytermsfor costs of adjustment. it is

(2) log y =log1f(K,Lt, Nt, Ht, S, Kt_t_l, L_q1L1

Nt -_1_1 Ht-H1,St -St1)]

= + + a0alog(K)aLlog(L) +aNlog(N) +aHb0g0)+ a5log(S)

- - - (l/)[g(KdK1)2 +g(L

- - - + gNA(N - qN )2 +g,(H H )2 +g(SS )2] . L—l1—i nfl L. L-L - [(K_1)(L_q1L1) +g(K_dKi)(H_H i

÷g5(N_q iNi)(S_Si) +g(H _Ht i)(S _S1)] f +a1t+v

Terms in levels are the standard, Cobb-Douglas, components. Those in changes represent the output foregone when the levels of factors are varied. The parameter d is the survival rate of capital (one minus the depreciation rate); the variable is the survival rate of labor (one minus the quit rate). In a related model [Shapiro (1984)], I find that the estimated adjustment cost parameters are insensitive to choosing different values of the elasticity of substitution in a constant elasticity of substitution specification. The cost of adjustment determines the dvnathcs of the model,

Hence, the Cobb-Douglas specification appears to be a reasonable, simplify- ing specialization in this class of applications.

Consider the effect on output of varyingthe work week of capital, 11

S .VaryingSt may involve costs of adjustment. As with hours, H.t

I expect the cost of adjusting the work week of capital to be small. Chang—

ing St ,ceterisparibus, involves a rescheduling of production, which

should be easy to do relative to changing the level of inputs.

Now consider the effect of the level of the work week of capital on

output. Suppose that the production function exhibits constant returns

to scale so that doubling capital and workers holding work week constant

doubles output. This implies the parameter restriction aK +aL aN =1 Suppose the firm wants to replicate the output produced by the plant not by doubling Kt ,L, and Nt , butby doubling the work week of capital keeping the capital-to-labor ratio constant. That is, a second shift

is added that replicates the first shift. To double output, the firm needs to double the labor input but only change the work week of capital holding capital stock fixed. Presuming labor is as productive at night as during the day, which is likely to be true as a first approximation, this scheme should just double output. This discussion abstracts from the adjustment costs.

This example illustrates a natural restriction on the parameters

, , and andS while hold- aK , aL aN aS . Ifdoubling L ,Nt, = ing Kt (and Ht )constantdoubles output, and aK + + 1 , aL aN then aK equals aS .Sucha result is highly intuitive. Extending the work week of capital creates capital services in exactly the same way as adding a new piece of equipment does. Therefore, Kt and S should have the same output elasticities. One would expect them to have very different costs of adjustment. In particular, one would expect the output foregone by adjusting the physical quantity of an input to far exceed the cost of changing the length of time that it operates. 12

Thecosts of increasing Kt and St are, however, very different.

In particular, increasing K involves the purchase of additional equip-

ment. How the cost of extending S is measured effects critically the

estimate of a .Idiscuss issues of measuring factor cost in the next

section.

3.2. Input Costs

The labor costs of a firm are a function of the number of employees, the number of hours they work, and the degree to which they work late shifts.

That is, in addition to the premium for overtime there is a premium for hours worked on late shifts.

I assume that non-production workers are paid a fixed amount. The wage bill of production workers is given by the expression

+ (3) wLH =wLt1J[w0 +.5w1(Hr/Ho) w2(SfI10)] + wherew is the average real wage, w the straight-time wage, H0 the average of Ht ,andv an error term. The term in S captures the premiumfor work at hours other than the standard shift. It ispaid when thework week of capital exceeds the work week of the typical worker.

The overtime premium, w1 ,istypically 0.5. Information on the reported premium for work on late shifts is available in the Area Wage

Survey. For 1973 through 1975, the average percentage pay differential for work on the second shift is 7.8 percent and for work on the third shift is 10.3 percent (see BLS Bulletin 1850-89, Area Wage Survey Summaries,

1975, p. 101).

The low value of the late-shift premium, w2 ,whichI confirm in the time—series estimates, poses theoretical difficulties for the model and estimates of the work week of capital. The premium is very small (corn- 13

pared to the overtime premium, for example). If w2 as directly estimated

represents the true marginal labor cost of extending the work week of cap-

ital, one is confronted with the puzzle discussed in the introduction:

Why is the work week of capital so low? Given the low incremental labor

cost, why is late shift work so rare? Put another way, if costs are indeed

this low, then labor productivity on late shifts must be very low. Based

on the low figure for w2 ,theestimate of a5 is substantially lower

than that of aK

Consider another alternative explanation for the low value of

Firms in manufacturing operating late shifts typically rotate such work

using a schedule that is essentially fixed among their existing work force.

If late work is expected and rotation is customary, most of the premium

needed to get workers to umdertake it may be built into the average wage

rather than explicitly made a function of late work. In this case, would be substantially underestimated. There is evidence in British data

that such practice is customary [see ?farris (1964, p. 137)].

Note that a similar argument cannot be made for overtime. The Fair LaborStandards Act requires that a standard premium be paid for overtime hours so these payments cannot be averaged into the straight-time wage.

Hence,the measured data for overtime, unlike those for late shifts, more closely approximate the marginal cost to a firm. There are no such legal restrictions on shift premium.

The proposed solution to the puzzle of why there is so little shift work given that the average cost of labor does not increase much on late shifts is that the average cost does not replicate the marginal cost. I posit that the true marginal cost may be better proxied by the overtime premium. Alternatively, one can ask what is the estimate of marginal cost given the theoretical condition that aK equala .Inthe next section, 14

I discuss the initlications of these considerations.

In addition to there being variable cost for production workers, there is fixed cost of employing both production workers and non-production workers. These are any payments not a function of hours worked, which, in the case of non-production workers, are all the costs. The fixed compensation cost of a production worker is denoted s ;the compensation of a non- production worker is denoted as

The real purchase price of capital is given by

- - (4) P tPVCCAt ITC) where p is the real, after-tax purchase price of capital, Pt the real before-tax price, t the corporate tax rate, PVCCA the present value of depreciation allowances, and ITCt the investment tax credit rate.

I assume that the survival rate of capital, d ,isa constant. Given that the intensity of use of capital is variable in the model this assumption may be controversial. The advantage of making the assumption is that it makes the capital stock easily measured. The disadvantage is that it may lose one of the substantial costs of increasing the work week of capital. The construction of these data is discussed in the Appendix.

3.3. The Firm's Decision Problem

The firm maximizestheexpected present discounted value of cash flow. The expected value of real, discounted, after-tax cash flow is given by 15

(5) EtR.{f(K.,Lt+j, Nt+±, Ht+j, Kj - Lt+j- Nt+± -

- — t+i t+i—l' t+i -t+i-1 t+i K -pt+i(Kt+_dK+i)

- + [w+.Lt+1Ht+.(wo.5wi(H./Ho)+w2(S ./H0)) + L N +s .L +s .N .jLlt .)J t+i t+]. t+i t+i t+i where Et denotes expectation taken conditional on information known to

the firm at time t . Themulti-period discount rate, , istime varying and random. It is defined by R+ = 1 where p is j=l• t+j—l the investorts required rate of return.8 In the following, the discount rate from time t to t+1 is denoted as r = Pt For the firm to be acting optimally, the first-order conditions of the problem must hold. Differentiating (5) with respect to capital,production workers, non-production workers, hours of production workers, and the work week of capital yields the five following stochastic first-order conditions:

(6a) E{IaK/K — - dKi)-g(L- - g(N-q1N1)

- - - - - + - g(HHi)gKs(StS1)]y(lt)[g(K1dK)

+ — + + - g(L1qL) (N41 -qN)g(H41Ht) + - S)]y+i(l-t+i)dr-p+drp+1}=0 16

(6b) E{[aL/L _g(K _dKi) -g(L_q iLt i -g(N_q1N1) -g(H-Hti)- - 1fl(l-t) +1+i — dKt)+g(L1-qL)+ - qN) _tK +g(H1 _H) +g5(S1 _Sfly+i(1 t+1)qr

- = [wH(wo+.5wi(H/Ho)+w9(S/Ho)) +s](1 -t)} 0

(6c) E{[aN/N _g(K. _dKi) -g(L_q 1Li) _g(N _q1N1)

- - g(H_Hi) -g(S_S ifly(l -t) + dK) +g(L1_qL) +g(N1 -qN)+g(H1 _H) -t)) =0

(ôd) E{{a/H -g(K-dKi) -g(L_qiLi) _g(N _q1N1)

- - + - g(H -H )- HH t t-1 - S1)]y(lt)[g(K1dK)

+ - - + - g(L41qL)+g(N1qN)g(FI1Ht) +g(S1 _S)]y+i(l _t1)qr _wL[wi(H/Ho)](1 -t)} =0

- (6e) Efla/S - - dK1)-g(L-q1L1)-g5(N N)

- - - - + _Hi)g(SS1)]y(1t)[g(K1dKt)

+ - + - + - g(L1qL)g(N1qN)g(H1Ht) +g5(S1 _S)]y+1(1 -t )r+1 -w Lt Httt (w 2 /H 0 )(1_tK)t =0 where

= Lt — f(K,Lt,Nt,H,S,Kt -dKtl, N_q1N1.Ht_Htl,S_Si) 17

Thefinal equation of the model is the expression for the wage bill of production workers (3). Iestimate equation (3) and the five first-order conditions (6) on data for U.S. manufacturing.The dataare described in the

Appendix.

4. Results

Estimation is carried out using non-linear three stage least squares.

Realized values replace the conditional expectations. The error terms in the resulting equations are strictly expectational (except for the measurement errors v in equation (3)). In particular note that the productivity shock v from equation (2) does not enter the system (6). That is, the shock is embodied by the output data. The shock does not appear even if it is serially correlated.9

Data known at time t are valid instrumental variables. The instruments used are the level and logs of the factors, Kt ,Lt.Nt and Ht , andSt , thelevel of the factor prices, w ,

, thetax rate, t ,survivalrate of labor, , therate of return, ri , anda constant and trend. Only the lagged rate of return is a valid instrument because rt is not known until the beginning of period t+l .Althoughoutput appears in the equations, it, like the factors, is endogenous. Output is not used as an instrumental variable.

Rotemberg (1984) shows that if the overidentifying restrictions of the model do not hold exactly what different lists of instruments can yield differ- ing estimates. I discuss the importance of this issue in Section 5.1.

Table I gives the estimates of the parameters of the firm's decision problem. Column (i) gives estimates where there are no interrelated adjustment costs; column (ii) gives them for where there are no adjustment costs for the work weeks of capital and labor, Ht and S. ,butwhere the stock of capital, production workers, and non-production workers, Kt 18

and , aresubject to interrelated adjustment costs. All other

interrelated adjustment costs always differ insignificantly and economic-

ally unimportantly from zero.

Consider the estimates of the parameters of equation (3) relating

hours and shift work to wages. The estimate of the overtime premium, w1 is 0.42 in column (i) and 0.50 in column (ii). The estimate in column (ii)

is exactly the theoretical value; the estimate in column (i) is close to

it. Hence, the estimate of the function for the wage bill is broadly con-

sistent with the stylized facts about overtime pay.

As anticipated in the theoretical discussion, the estimated premium

for work on the late shift, w2 ,issmall. The estimated premium of 0.05

in either column (1) or (ii) is consistent with the observation from the

Area Wage Survey discussed above. The time series evidence does not rely

on the information about the wage premium given in the Area Wage Survey.

Both the evidence and the time series evidence point to a late shift prem-

ium substantially below that of the overtime premium. I discuss below the

implications of the estimate of w2 for the estimate of the output elas-

ticity of the work week •of capital.

Consider now the estimates of the output elasticities. The elastic-

ities for production workers,aL , andnon-production workers, aN

and the implied elasticity for capital, aK , areconsistent with their shares in national income. The elasticity of hours of production workers,

Ht , isonly slightly larger than that of the stock of production workers,

Lt . Ifthey were equal, production workers and their hours would be perfect substitutes!0 In that case, the level of labor input could be treated as man-hours rather than workers and hours separately. The adjustment costs, which depend on the changes in inputs, and the contribution to the wage bill, are, of course, still very different for hours and workers. 19

The output elasticity of the work week of capital,a ,isesti- mated to be only 0.026. Recall that the theoretical value ofa is equal

to that of aK ,whichis plausibly estimated to be about one-quarter.

Hence, if the estimate of the shift premium is taken from the data on the

average wage, the productivity of shift work is estimated to be very low.

The only way for the model to be consistent with that data is for the implied productivity of lengthening the work week of capital to be very low.

This finding is consistent with the observed low apparent productivity of shift work found in studies discussed in footnote 2.

As argued above, the data on average wages may be misleading about

the marginal cost of work on late shifts. Specifically, if firms and workers

have an implicit contract to share work on late shifts equallyamong workers

and if the pattern of such work is reasonably predictable, the straight-

time wage might average a low wage for work during the day and a highwage

for work during the night. In that case, the average wage is apoor mdi- 11 cator of the marginal cost of increasing shift work to the company.

Consider the coefficients measuring the lost output from changing the amount of an input. Column (i) of Table I gives estimates without interrelated adjustment costs. Colun (ii) gives estimates with interrelated adjustment costs for capital, production workers, and non-production workers. All other interrelated adjustment costs are insignificant and insubstantial. Indeed, the adjustment costs for hours and shifts are likewise insignificant and economically unimportant so they are left out of column (ii).

It is difficult to evaluate the size of the adjustment costs based on the parameters alone. It is informative to consider the marginal adjustment costs that they imply. Consider first the costs of adjustment in the equation for capital. The estimate of g of 0.0008 in column (ii) im- 20 pliesan adjustment cost of 5.0 percent of the cost of investment. That is, for an average amount of investment, if there is no gross change in the amount of labor, output is reduced by 5.0 percent of amount of the investment. Average investment in the sample is 9.6 and average output is

62 measured in billions of 1972 dollars at quarterly rate. Therefore, the foregone output caused by adding the average amount of capital is less than one percent of average output. In particular, it is substantially smaller thanother estimates ISummers (1981) and Heese (1980) ,forexample]. Pindyck and Rotemberg (1983) ,onthe other hand, share my finding of moderate ad- justmentcosts. The interrelated adjustment costs for capital and production workers and non-production workers are small: they are an order of magnitude smaller than the own-adjustment cost of capital. The sign of g is negative, so the cost of adjusting capital is reduced when non-production workers are also being adjusted. That is, changing the stock of capital is facil- itated by adding non-production workers.

It is more difficult to evaluate the adjustment costs of labor because although the stock is being adjusted, labor, unlike capital, is paid as a flow. In any case, one can compare the marginal adjustment costs to the flow cost of labor. In the case of production workers the adjustment costsare all small. Consider the estimate of 0.0403 for g in (ii), that is, the adjustment cost for non-production workers. It is substantial: the marginal adjustment cost for the typical change in non-production workers isabout3percentof outputperquarter. Non-production workers receive about one quarter of output.

Insummary,the adjustment costs of both the work week of capital and of labor are small. This result is not surprising given the relative ease of adjusting the schedule of production relative to adjusting the stock 21 ofan input. The cost of adjusting production workers is also small. The cost ofadjusting capital is significant but substantially smaller than found in other studies. Thus, the estimate iuqlies quicker adjustment of thecapital stock than others find.

5. Extensions

5.1. Failure of the Overidentifying Restrictions

The J statistic given at the bottom of Table I is a test of the overidentifying restrictions of the model. It is distributed as chi.-squared with degrees of freedom equal to number of instruments minus number of parm- eters. There are 114 instruments (19 instruments times 6 equations) so the overidentifying restrictions are soundly rejected in all the specifi- cations.

Rotemberg (1984) suggests that if the overidentifying restrictions of such a model are rejected, different instrument lists can yield widely different estimates. Specifically, if the weighting of the instruments is arbitrary, the estimates can have any probability limit. Note that the weighting scheme is determined here not arbitrarily, but optimally by three- stage least squares.

In order to evaluate the problem raised by Rotemberg's work,

I present estimates of the model with alternative instrument lists.

The first uses just price data; the second uses just quantity data. This division is economically as well as statistically meaningful. That is, the stochastic properties of prices and quantities are very different. In particular, factor quantities have high variance and are strongly pro-cyclic; factor prices vary little and are only weakly cyclic.

Estimates of the model with the two instruments lists are presented in columns (ii) and (iii) of Table II. The exact lists are given at the bottom of Table II. Both lists yield overidentification, but to a lesser degree than by taking the union of the lists. Column (i) replicates column

(ii) of Table T for comparison. In columns (ii) and (iii), the parameters

and w2 are constrained to have the same value as in column (1). The estimates of w1 and w2 ,theparameters of the equation for the wage bill (3), require both price and quantity data to be identified. Hence, they are left out of the experiment.

The estimates in column (ii) and column (iii) of Table II also strongly reject the overidentifying restrIctions. Therefore, one cannot infer either that it is only the price data or only the quantity data that are invalid

instruments. An economically meaningful way to evaluate the problem is to see how much the estimates change given the rejection.

The estimate of aL ,theproduction workers' output elasticity,

changes little over the choice of instruments. Likewise, the coefficient

aH is stable. The coefficient aN ,theoutput elasticity for non- production workers, and consequently capitaPs implied output elasticity,

does change substantially. The implied aK is column (ii) is 0.18 which is somewhat lower than conventional estimates. The coefficients in the adjustment cost change more, but they are less precisely estimated in the first instance. The most serious change occurs in g , wherethe estimate not using price data implies a very high adjustment

Cost.

Errors in variables can be analyzed as a failure of the overidentifying restrictions. The work week of capital contains a measurement error because of the distribution of annual data to quarterly frequency.

The importance of this problem can be evaluated by considering estimates where St is excluded from the instrument list. These estimates may be consistent even if St is measured with error. In column (Iv) of Table 2, I present estimates of the model whereSt and its log are excluded from

the instrument list. The estimates are virtually identical to those in

colunni (i). Hence, the measurement error does not have a quantitatively

important affect on the results.

The results of the analysis of varying the instruments is indecisive.

In particular, it does not isolate the invalid instruments as either belong-

ing to the price or quantity subset. With either set of instruments, about

half the coefficients are remarkably stable. The other half change by

amounts that would importantly affect the dynamics implied by the model,

but they never change so much as to take on values that could be excluded

a priori. Moreover, it is a minor victory that I am able to arrive at a

not entirely implausible set of estimates with the price data alone as in-

struments even though they vary much less than the quantities. In any case,

one is less confident about the point estimates of the parameters that change

importantly.

The rejection of the overidentifying restrictions indicates either

the theory is false or that some of the strong auxiliaryassumptions are

false. These assumptions include a Cobb-Douglas production function and

the presence of only one lag in the cost of adjustment term. Theseassump-

tions could be relaxed by less parsimonious paranieterizations. Acost of

less parsimony is increased difficulty in interpreting the estimatedpararn- eters. On the other hand, a less parsimonious model is much more likely to fail to reject the overidentifying restrictions. Seeking lessparsiinon- ious models that fail to reject overidentifying restrictions isnot necessarily a promising research strategy. Tests of such models are likely to lackpower. 24

5.2. Aggregation Problem

Problems with using aggregate data are pervasive in macroeconomics.

There is typically a tension between the theoretical discussion and the

empirical implementation. The aggregation problem is, perhaps, more severe

or more obvious, in the application to shift work. Specifically, finns

may be heterogeneous in their policies toward shift work. This section

evaluates the consequences of this aggregation problem for the analysis

in this paper.

Suppose that there are two types of firms, one that always works

only one shift and one that has varying shifts. Let S1 be the constant work week of the first type of firm and S2 be the work week of the second

type of firm. I assume that the fraction of type one firms is a constant,

b . Then

S = 4 (7) bS1 (l-b)S2

All the variation in S comes from the second type of firm. Substituting

(7) into the objective function (5) and differentiating with respect to

the choice variable, S2 ,yieldsthe following modified first-order condition for S

(6e') a5(1_b)Y/S =WtLtHtW2/HO

In light of the estimates of the adjustment cost coefficients, I set them

to zero. The tax rate cancels so (6e') is the familiar marginal product

equals marginal cost condition. The estimates of a in Table 2 do not

allow b to be separately identified. That b is large is another pos-

sible explanation for why a is estimated to be so much lower than aK

when w2 is freely estimated. If many firms, for whatever reason, never 25

work late shifts, then the interpretation that shift work is unproductive

seems to be correct. In any case, b can be regarded as a normalization

of equation (6e') because a5 is a free parameter. Therefore, the reasons

discussed in footnote 11, whether there is an aggregation problem of this

type affects the interpretation of the coefficients but does not change

the dynamics iilied by the estimated model.

6. Dynamics

In this section, I explore the dynamics implied by the estimates of the model. In particular, I examine the effect of changes in the price of capital and investors' required rate of return on the demand for capital and its work week. The estimates have broader application, but the aim here is to study the demand for capital. Estimating the first-order conditions of the model allows more realistic parameterizations than estimating the

solution directly. The model cannot, however, be solved analytically.

To study the dynamics, I linearize the first-order conditions to study the

response of the model to small changes in the cost of capital.

The exact procedure I follow is (a) linearize the estimates of equations (6) and (3) from Table I, column (ii), about the sample average values for the variables;12 (b) put the linearized system in the canonical form of Blanchard and Kahn (1980) and solve it; and (c) examining how the factor 13 demands change when the price affecting them are changed. Linearizing these equations is unlikely to be misleading. First, the estimated cost of adjustment parameters carry implications for thedynamics.Thesimula- tions serve to illustrate what the estimates already imply. Second, the cost of adjustment terms are linear except for multiplication by the level of other variables. These terms govern the dynamics of the simulation. 26

Abel and Blanchard (1983) show linearizing such functional forms is a good approximation.

Some assumption must be made to close the model. In particular, though the representative firm in the manufacturing sector is a price-taker in the labor market, the sector as a whole does not face elastic labor supply.

To the contrary, labor supply is highly inelastic, and hence bounds the response to a shock that could otherwise make the economy expand indefinitely.

I add a final relationship to the system to take into account inelastic labor supply. I assume that the labor supply curve has a constant elasticity of o.i.14 The equation that labor supply equals labor demand is solved with equations (6) and (3).

I consider the effect of changing both the purchase price of capital and the investors? required rate of return on the demand for capital and itswork week.I consider decreases in these two components of the iurplied cost of capital, but the results are exactly reversed for increases. A decreasein the required rate of return of investors increases the discount rate and hence increases the value of the firm. The discount rate, r enters the del highly non-linearly. As I outline above, this problem is overcome by linearizing the system. Bernanke (1983) uses a similar procedure.

Figures 1 through 4 give the response of the capital stock and its work week to different changes in the price of capital and the rate of return. The changes can be thought as changes in tax rates. A decrease in the purchase price of capital can arise when the investment tax credit is increased or depreciation allowances are liberalized. A decrease in investors' required rate of return may occur when the tax rate on investment income isreduced. 27

In Figure 1, I present the change in the demand for capital and its

work week for a permanent, ten percent reduction in the cost of capital.

The reduction is unexpected; once it occurs, it is expected to, and indeed

does, last indefinitely. The first column gives the change in the price.

The second and third columns give the percent change in the stock and work

week of capital. The fourth and fifth columns give the levels of the stock

and work week (deviation from steady state). In the long run, the capital stockincreases by 3.1 percent and the work week of capital increases by 9.7 percent iiying respective elasticities of about one-third and one. The

capital stock is costly to adjust so itrespondsonly gradually to the change inprice. Half the adjustment has taken place after five quarters, which

is much more rapid than found in other studies. In a similar experiment,

Summers (1981) finds half the adjustment taking twenty years. The magnitudes

of the long run changes are comparable with those Summers estimates.'5

The work week of capital is costless to adjust. Indeed, it overshoots

its steady state value by 17 percent in quarter 3. It then gradually falls

to its steady state value. For the entire period of adjustment, the firm has a lower capital stock than it desires given the new factor prices.

Hence, it correspondingly increases the utilization of its stock by expand- ing its work week. Note that the figures give the response of demand for capital and its work week to changes in prices. The response of equilibrium quantities will be smaller and depend on elasticities of supply. In particular, the interest rate may rise when demand for capital is stimulated by policies such as the investment tax credit. Noreover, the premium for work on late shifts may rise in the face of large increases in demand by firmsfor night work. 28

In Figure 2, I present the response of capital to the ten percent decrease in its cost that is expected in quarter I to be permanent. In quarter 9, the price of capital unexpectedly increases to its original level. Through quarter 8, the dynamics are exactly as in the first example. In quarter 9, when the firmsdiscoverthat the reduction is only temporary, they suddenly have capital substantially above the steady state value of zero. Because of the ad-ustment cost, the stock only gradually returns to its steady state value. On the other hand, the rate of utilization is free to adjust and therefore overshoots to offset the capital stock being above its steady-state value. The work week approaches the steady state from below.

Figure 3givesthe response of capital to a permanent decrease in investorst required rate of return. It has a large effect on the demand for capital because it increases the present discounted value of cash flow as well as reducing the implied rental rate on capital. The steady state increase in the capital stock from a ten percent reduction in investors' required rate of return is 20.3 billion 1972 dollars, or ten percent of the average stock.16 Because of the linearity of the solution, the relative changes of the stock and the work week and the rates of adjustment are exactly as in Figure 1. Note that the increase of the capital stock would be smaller if the shock to the required rate of return were not permanent, or, 'of course, if the shock were smaller.

In a final experiment, I consider an expected, future, permanent reduction in the investors' required rate of return. But when the reduction is expected to take place, it is revealed that no reduction will take place. Hence, the interest rate never changes. These dynamics are displayed in Figure 4. Such a scenario could occur if the govermnent announces a tax cut on investors' income to take effect in 9 quarters, if 29

the firms believe the government, and if the government then reneges on its

commitment. This is an effective policy for temporarily increasing the

capital stock, but one which cannot be repeated often [see Fischer (1980)].

Future flows from the current capital stock are discounted at a lower rate,

so the stock increases. In quarter 9, the firms discover they have been

misled. They only gradually reduce their capital stock, but immediately

adjust its utilization. Indeed, the utilization rate overshoots its steady

state value and approaches it from below.

I find a potentially substantial effect of the real interest rate and

the purchase price of capital on the demand for capital. The standard view

is, however, that "The rental price of capital, a conglomeration of interest rate and tax variables, is not very useful in explaining quarterly data of business fixed investment in the United States over the past twenty-five years" fClark (1979, p. 104)] .Clarkdoes not conclude that the cost of capital is unimportant in theory, but that they do not vary enough given slow adjustment of the stock to be important in the data.

The findings of this paper provide important challenges to this standard view. First, I find a substantial effect of interest rates and factor prices on the demand for capital. Second, the adjustment costs I estimate are small enough so that the capital stock can respond somewhat to business-. cycle frequency changes in interest rates and prices. Third, lower interest rates increase the present discounted value of cash flow so future labor services, which will be associated with the capital services, are also more highly valued.

It is useful to consider why prices and interest rates are important in my model but not in non-structural reduced forms such as considered by

Clark. ISee also Bernanke (1983) for a related discussion.] My procedure 30

is as follows: I specify a structural model; I estimate the first-order conditions of the model with a parameterization that is too complicated to solve explicitly; and I linearize the estimated system to study its dynamics.

Clark estimates linear reduced forms. Although the linear reduced form may correspond to some structural model, estimating it may not obey the restrictions implied by the production function and by profit maximization

(in equation (6) for example))7 Consider equation (6a). The interest rate enters the implied rental cost of capital by multiplying p ,but also by multiplying the future value of the factors. The implied reduced form is highly non-linear in variables and parameters. It is easy to impose the appropriate restrictions by estimating the first-order condition.

By linearizing after estimating, I impose the restrictions, at least locally, when examining the effect of prices and interest rates on investment.

7. Conclusion

This paper provides the analytic mechanism and empirical evidence to analyze the dynamic demand for capital, labor, and their levels of utilization. The work week of capital measures the utilization of the stock.

It is constructed using data on the amount of shift work.

The productivity of extending the work week of capital must be low relative to the costs given how few late shifts are worked. The average wage premium for working on late shifts is low. This implies that the output elasticity will be estimated to be low.I consider two alternative explana- tions of these facts, both of which are consistent with the data, and both of which imply the same macroeconomic dynamics. The first is that the 31

marginal wage premium is much higher than the average wage premium. The

secondis that there is an aggregation problem if some firms have rules

against late shifts.

The estimated adjustment costs are also consistent with theory. The adjustment cost on capital is again estimated to be much lower than found in other studies, imtlying more plausible rates of response to stocks.

As with hours of production workers, the work week of capital is virtually costless to adjust. Thus, it provides a margin to easily absorb transitory shocks to product or factor prices.

Additionally, when the work week of capital is taken into account, the output elasticity of the work week of labor is not substantially larger than that of the stock of labor. Hence, the greater marginal product of hours (Ht) compared to workers (Lt) the work week of capital is excluded can be understood to arise because those extra hours imply a lengthening of the work week of capital, not because a current worker working an extra hour is more productive than a new worker. The estimates of (6) provide some justification for letting hours and workers enter multiplicatively in levels in the production function. On the other hand, their adjustment costs and contribution to labor's compensation are very different.

The work week of capital is essentially costless to adjust so it will respond immediately to shocks whereas the stock of capital will respond slowly. The plausibly estimated adjustment costs imply that the rate of adjustment is not instantaneous, but is much more rapid than as estimated in other studies. The work week of capital provides an extra margin along which to adjust. In the response to a shock, the work week of capital will overshoot to compensate for the slow adjustment of the stock. The ma2nitude of the overshooting is about twenty percent. 32

I find a large effect on the capital stock of changing interest rates

andfactorprices. These estimates provide a challenge to the standard view that prices are not important in determining demand for capital. The

long run elasticity of the capital stock with respect to the price of capital is estimated to be about thirty percent. The elasticity with respect to the required rate of return is about one. -D)

TABLE I

Estimateof First-order Conditions (6) and Equation for the Wage Bill (3) 1955 QIlI through 1980 QIlI

(i) (ii)

.76 .72 (.016) (.023) w .42 .50 (.030) (.044) .05 .05 (.001) (.001)

aL (.005) (.001)

aN .28 .27 (.004) (.004)

aH .48 .48 (.008) (.011) .027 a5 .026 (.001) (.001) .0005 .0008 (.0004) (.0006) * -.0009 (.0009) .1275 .0403 (.0296) (.0302) —.0001 (.0001) * ss .0016 (.0005) -.0020 (.0019) .0149 (.0036)

.1 358 362

Standarderrors in parentheses. test of overidentifying restrictions [see Hansen (1982)].

*indicatesmagnitude of estimate <.00005. 34

TABLE II

Estimate of First-order Conditions (6) and Equation for the Wage Bill (3) 1955 QIlI through 1980 QIlI Alternative Instrument Lists

(i) (ii) (iii) (iv) Instrument list: AB A B C wo .72 .72 .72 .72 (.023) (.0004) (.0004) (.0004)

Wi .50 .50 .50 .50 (.044) W .05 .05 .05 .05 (.001) aL .47 .47 .47 .47 (.001) (.008) (.005) (.005) .35 .27 .27 aN .27 (.004) (.020) (.004) (.004) a .48 .48 .47 .48 (.011) (.007) (.007) (.007) a5 .026 .026 .025 .026 (.001) (.0004) (.0004) (.0004) KK .0008 .0005 .0026 .0008 (.0006) (.0008) (.0008) (.0007) -.0009 —.0046 .0009 -.0007 (.0009) (.0033) (.0014) (.0009) .0403 .7641 .0873 .0448 (.0302) (.3260) (.0507) (.0310) .0016 .0023 .0034 .0018 (.0005) (.0011) (.0007) (.0005) -.0020 -.0146 -.0013 -.0015 (.0019) (.0128) (.0029) (.0019) .0149 .0155 LN .0382 .0169 (.0036) (.0244) (.0063) (.0036) J 362 176 262 317 k 114 48 72 102

Standard errors in parentheses. J test of overidentifying restrictions [see Hansen (1982)], which is distributed as chi-squared with k degrees of freedom where k is the number of instruments with time equations. In columns Cii), (iii), and (iv), w1 and w2 are constrained (see text). *indicatesmagnitude of estimate <.00005. Ins truinentlists: K K L N A constant,•.trend,t ,w,p ,r,s ,s Bconstant,trend, K, L, N, H, S, log(K), log(L), log(N), log(H), log(S) C constant, trend, tK, w, K, r, 5L, 5N q, K, L, N, H, log(K), log(L), log(N), log(H) (that is, all excluding S and log(S)). 35

Figure 1

Response of Capital to a Permanent,UnexpectedTen Percent Decrease in the Price of Capital

0.0 0 10 20 30 40 50 60 quotiers

Note: Stock measured in billions of 1972 dollars.Work week measured in hours. 36

Figure 2

Response of Capital to an Unexpected Ten Percent Decrease in Its Price Expected to be Permanent But Actually Lasting Eight Quarters

7.0

5.0

3.0

1.0

-1.0 0 20 30 40 50 60 quarters

Note: Stock measured in billions of 1972 dollars. Work week measured in hours. 37

Figure 3

Responseof Capital to a Permanent, Unexpected Ten Percent Decrease in Investors' Required Rate of Return

0 10 20 30 40 50 60 quarters

Note: Stock measured in billions of 1972 dollars. Work week measured in hours. 38

Figure 4

Response of Capital to a Future, Permanent Ten Percent Decrease in Investors' Required Rate of Return When the Decrease Does Not 0 c cur

7.5

5.0

2.5

-2.5 0 10 20 30 40 50 60 quarters

Note: Stock measured in billions of 1972 dollars. Work week measured in hours. APPENDIX

A. Capacity Utilization Data

This Appendix discusses how published data on capacity utilization do

not represent directly data on capital utilization. Ideally, one would have

an independent measure of the rate of utilization of each of labor and cap-

ital. At any point in time, the firm has four choices, how much labor and capital to hire or buy, and at what rate to use each. Because higher utilization has a cost, it does not always pay to have maximum utilization.

There are no published figures at the macroeconomic level for utilization of capital analogous to average hours of labor. An analogous figure would measure the physical utilization of the capital. For a machine it could measure the rate of speed of operation or percentage of the time it is in operation. For a structure, it could measure the percentage of the time it is in use. The published figures for capacity utilization reflect the ratio of output to "potential output" rather than the actual utilization of capital. One could imagine using energy consumption as a proxy for capital utilization. Such a procedure again asks how much of an alternative factor is utilized.

The Fed constructs its index of capacity utilization by dividing its index of industrial production by an index of capacity measured in units of output. [See the Federal Reserve System, Measures of Capacity and

Capacity Utilization (1978) for details.] It bases the index of capacity on McGraw-Hill surveys and the BEA's data on capital stock. The capacity figures are available only annually; the Fed interpolates the within-year figures. Therefore, most of the variation in the Fed's index of capacity utilization is caused by variation in output. Indeed, the concept being 40

measured is close to output divided by potential or trend output.

The BEA constructs its series for capacity utilization from a quarterly survey of firms. In the survey's questionnaires the BEA fails to define what it means by capacity utilization but it "believe[s] that most respondents use a measure of 'maximum practical capacity. " [SeeBureau of Economic Analysis, Handbook of Cyclical Indicators (1977, p. 25), emphasis added. ]TheBEA defines maximum practical capacity as output where a standard work-week's of labor to be supplied.

Variations in both measures of capacity utilization come mainly from the amount of labor input. Therefore, the choice of measured capacity utilization is not independent from the choice of other inputs.

B. Data for the Estimates

The data are quarterly for manufacturing from 1955 through 1980.

The output data are the Federal Reserve Board's quarterly index of output in manufacturing scaled so it equals annual NIPA output in 1967.

The quarterly data for investment is from the Survey of Current Business.

Structures and equipment are aggregated. I construct the capital stock data (Kt) using a fixed depreciation rate of 0.0175 per quarter and a benchmark net capital stock of 311.8 billion 1972 dollars at the end of

1981 (see the Survey of Current Business (October 1982, p. 33)). This depreciation rate is also imposed in estimating the Euler equations.

The data for employment (Lt, N) ,wages,the quit rate, and hours are from the BLS establishment survey. The wage (wt) series is the average straight-time rate per hour for the production workers. The data on average hourly earnings (w) are also used in equation (3). The hours

(Hr) series is total average weekly hours. Hours are multiplied by the 41

number of weeks in the quarter in the marginal cost expressions in the Euler equations to express cash flow at quarterly rate. To construct the fixed cost of employing a worker s- and sN ),Idivide the compensation minus wages, salaries, and contribution to social insurance into the number of workers using annual national income and product accounts data. These costs include employer contributions to pensions and health insurance. The annual

NIPA figures are interpolated.

The expression for the price of capital is given in equation (4).

The purchase price is the implicit deflator from the BEA. The quarterly series for the present value of depreciation allowances (PVCCAt) and the investment tax credit (ITCt) are those computed by Data Resources

Inc. The present value of depreciation allowances are computed using the technique outlined by Jorgenson and Sullivan (1981). Expected tax savings assuming current law remains unchanged are discounted by using the firm structure of interest rates.I estimate the required rate of return with the after-tax, real return on three-month Treasury bills plus a constant risk premiuiu of two percent per quarter. I calculate the premium by taking a weighted average of thereturnin excess of the return on Treasury bills of the stock market and of corporate bonds. The weight for equity is 0.8.

The excess return of the stock market is 6,7 percent; the excess return of corporate bonds is 0.6. ISee Ibbotson and Sinquefeld (1982, p. 15) .1There- fore, the premium is about eight percent at annual rate or about two percent at quarterly rate.

Construction of the work week of capital is discussed in detail in the text. The data are given in Table A-i. 42

TABLE A-i

The Work Week of Capital: St United States Manufacturing Industry (hours)

QI Qil Qill QIV

1952 54. 56.1 55.9 55.7 1953 55.1 55.4 55.4 55.3 1954 53.4 49.9 49.1 52.7 1955 54.3 54.6 54.7 54.8 1956 55.1 55.2 55.2 55.3 1957 54.8 54.9 54.6 52.8 1958 54.0 52.8 52.9 53.0 1959 55.3 54.4 53.5 53.3 1960 55.0 53.9 53.5 52.8 1961 53.6 53.6 53.8 53.9 1962 55.1 54.8 54.8 54.8 1963 55.0 55.1 55.1 55.0 1964 55.2 55.2 55.4 55.5 1965 56.9 55.5 55.9 56.3 1966 57.4 57.0 57.3 57.5 1967 57.8 55.6 56.3 56.5 1968 58.9 56.3 57.1 57.8 1969 54.6 54.8 55.4 55.0 1970 54.4 53.5 53.8 53.2 1971 53.7 53.8 54.3 54.8 1972 55.4 56.0 55.6 56.1 1973 56.0 56.2 56.2 56.3 1974 56.0 55.0 56.0 54.6 1975 54.1 54.3 55.1 55.7 1976 55.2 54.9 55.0 54.9 1977 54.9 55.7 55.7 55.9 1978 55.0 56.1 56.2 56.4 1979 56.5 55.0 55.6 55.4 1980 55.5 54.0 53.7 54.7 1981 54.4 54.6 54.1 53.2 1982 53.2 53.8 53.5 53.4 43

FOOTNOTES

1Thispaper is a revision of the second chapter of my 1984 M.I.T. Ph.D dis-

sertation. I am very grateful to my committee, Stanley Fischer, Jerry

Hausman, and Olivier Blanchard, and to Andrew Abel, Ernst Berndt, Zvi

Griliches, N. Gregory Mankiw, James Poterba, David Romer, Julio Rotemberg,

and Lawrence Suimners for their extensive comments and discussion. I grate-

fully acknowledge the financial support of the National Science Foundation andthe Social Science Research Council Subcommittee on Monetary Research.

has longbeen observed that most capital in most countries is idle most of the time (.see Foss (1963, 1981) and Betancourt and Clague (1981)). For example, the average work week of capital for United States manufacturing industries is 55 hours according to the series to be discussed below. This low level of utilization is common across countries. Moreover, businesses typically plan investment so that most capital is idle most of the time

(see Marris (1964, 1970)). That is, many firms plan to operate only one shift. Given that the productivity of capital is not constrained by pref- erence, custom, law, or biology as is labor, it seems surprising that the work week of capital only slightly exceeds that of labor. The biological and social constraints on shift work are discussed in Mott, etal. (1965),

Maurice (1975), and Hedges and Sekscenski (1979). Mott, etal. provide survey evidence on attitudes of workers and their families toward work on late shifts. Work on late shifts yields substantial disutility beyond work in the day, but the added disutility does not appear to be enough to explain the low level of shift work. Maurice provides a summary of law about shift work for a sample of developed countries. Hedges and Sekscenski survey the 44

biological literature on the physiological effects of work on late shifts.

Winston (1974, 1982) surveys the theoretical literature on the utilization puzzle. Betancourt and Clague study the choice of shift work in a static context.

31f there is only one shift then average hours of workers would also be the average hours of capital.

4The sources of the annual data on the distribution of shift work is the

Bureau of Labor Statistics, Area Wage Surveys: Metropolitan Areas, United

States and Regional Summaries, various years. [The BLS report numbers are

1660—92, 1685—92, 1725-96, 1775-98, 1795-29, 1850—89, 1900—82, and 1950-77,

Table B-i.] I am grateful to the BLS for providing me with unpublished data for years after 1978. Murray Foss provided me with part of an unpub- fished monograph which alerted me to these data.

51n the estimates of the model itself, the data are not detrended.

6Feldstein and Foot (1971) and Eisner (1978) include capacity utilization in a regression to explain investment. In their models, firms are quantity constrained, so capacity utilization is a proxy for the shadow profitability of investment. A quantity-constrained firm with low utilization has little incentive, to add capacity.

7Shapiro (1984) gives more detailed consideration of a related model but where capital is utilized at a constant rate given labor input. 45

8i estimate the required rate of return withtheafter-tax, real return on

three-month Treasury bills plus a constant risk premium of two percent per

quarter. See Appendix for details. An instrumental variables procedure

allows consistent estimation even though the rate of return is uncertain

and may, in general, covary with the other variables in equation (5).

9Use ofoutput data makes theproductivityshock observable (abstracting from the sampling error in the estimated parameters). Hence, these esti-

mates do not become unidentified in the presence of productivity shocks as

discussed by Garber and King (1983). In their paper, the shocks are unob-

servable. Of course, identification of this model, as with all models,

depends on assumptions on the error terms. In any case, the approach in

this paper of incorporating a particular productivity shock seems a substan-

tial improvement over the standard practice of assuming that there are no

shocks at all.

10 If shifts are omitted from the analysis, theoutput elasticity of hours will be substantially greater than that of production workers because the hours of workers are then in part proxying for the hours ofcapital.

Consider rescaling the shift work equation (6e) by multiplying it by a constant. Note that the estimated coefficients for the cost of adjustment

' , andso on) are essentially zero. Hence, multiplying (6e) by a constant will simply increase a and w2 by the sane proportion.

Suppose we wished to constrain aS to equal the estimated aK ,which is about one-quarter. To achieve this, equation (6e) would be multiplied by about ten. In particular, the implied w2 would be about ten times 46

the estimated value of 0.05, that is 0.5. Hence, if we imDose that the

productivity of capital and shifts are the same, the implied shift premium

is exactly the overtime premium. Hence, one way of understanding the low propensity to use late shifts is that the actual cost of running late shifts is substantially higher than the data seems to ily. Note, however, that this issue only effects the interpretation of the coefficients. The dynamics studied below are unchanged by the scale of equation (6e).

12The average values needed to carry out the linearization are =62.5

= = = = = 4.8 , 40.2, 55.0 , 0.94 Kt =202.0, Lt 13.6 , N II St q. = = = d = =142.1 =1698.1' ' 0.68 , 1.75 0.9825 , s , s P w and t =0.49 . Tocalculate the intercept, a0 ,inthe production function (2) I set it so (2) holds given the average values and the parameter

estimates. I ignore technological progress so the simulations are devia-

tions from the steady state.

alternative would be to simulate the non-linear Euler equations. Given

that more assuniptions are needed to simulate the model than to estimate it

(see footnote 14) ,studyingthe linearized system is a cost effective way

to approximate its dynamic properties. The simulations are carried out

using a computer program written byJefferyZax and provided to me by

Olivier Blanchard. 47

14Hausman (1981) estimates theuncompensated wage elasticity of labor suDply

to be virtually nil for adult males, but substantially higher for females.

Ten percent in a compromise. I assume that non-production workers are in

elastic supply because the sector demands relatively few of them. Also,

implicit in the simulations is that the price of capital is given except

for the tax considerations and that capital is elastically supplied.

'5Suinmers studies the entire non-farm businesseconomy which is over four

times the size of manufacturing. His simulation in Table 6 (p. 37) yields

results substantially similar to those in this paper in the long run, but

with much slower adjustment to the steady state.

16This change in the capital stock is larger than Summer's(p. 109). He

assumes inelastic labor supply, which could partially account for the differ-

ences. It may also be the case that my linearization breaks down for changes

this large.

'71n that I reject the overidentifying restrictions my estimates are not

consistent. It is difficult to judge the consistency of estimates in

Clark'S tradition. 48

REFERENCES

Abel,A. B. (1979), Investment and the Value of Capital, Garland, New York.

______and0. J. Blanchard (1983), "The Present Value of Profits and

the Cyclical Movements of Investment," National Bureau of Economic

Research Working Paper No. 1122, Cambridge, MA. Forthcoming in

Econometrica.

Betancourt, R. R. and C. K. Clague (1981), Capital Utilization: A Theoretical

and Empirical Analysis, Cambridge University Press, Cambridge.

Bernanke, B. S. (1983), "The Determinants of Investment: Another Look,"

American Economic Review Papers and Proceedings, 73, 71-73.

Blanchard, 0. J. and C. H. Kahn (1980), "The Solution of Linear Difference

Models under Rational Expectations," Econometrica, 48, 1305-1311.

Board of Governors of the Federal Reserve System (1978),Federal Reserve

Measuresof Capacity and Capacity Utilization, Washington.

Bureau of Economic Analysis, U.S. Department of Commerce (1977), Handbook

of Cyclical Indicators, G.P.O., Washington.

Bureau of Labor Statistics, Area Wage Surveys: Metropolitan Areas, United

States and Regional Summaries, various numbers.

Chow, G. C. and A. Lin (1971), "Best Linear Unbiased Interpolation Distribu-

tion, and Extrapolation of Time Series by Related Series," Review of

Economics and Statistics, 53, 372-375.

Clark, P. K. (1979), "Investment in the 1970s: Theory, Performance, and

Prediction," Brookings Papers on Economic Activity, No. 1, 73-113.

Eisner, R. (1978), Factors in Business Investment, National Bureau of Economic

Research General Series No. 102, Ballinger, Cambridge, MA. 49

Feldstein, M. S. and D. K. Foot (1971), "The Other Half of Gross Investment:

Replacement and Modernization Expenditures," Review of Economics and

Statistics, 53, 49-58.

Fischer, S. (1980), "Dynamic Inconsistency, Cooperation and Benevolent Dis-

sembling Government," Journal of Economic Dyanmics and Control, 2, 93-107.

Foss, 1. F. (1963), "The Utilization of Capital Equipment: Postwar Compared

with Prewar," Survey of Current Business, 43, 8-16.

______(1981),Changes in the Workweek of Fixed Capital: U.S. Manufacturing,

1929 to 1976, American Enterprise Institute, Washington.

Garber, P. M. and R. G. King (1983) ,"DeepStructural Excavation? A Critique

of Euler Equation Methods," National Bureau of Economic Research Tech-

nical Working Paper No. 31, Cambridge, MA.

Hansen, L. P. (1982), "Large Sample Properties of Generalized Method of Moments

Estimators," Econometrica, 50, 1029-1054.

______andK. S. Singleton (1982), "Generalized Instrumental Variables

Estimation of Nonlinear Rational Expectations Models," Econometrica,

50, 1269-1286.

Hausman, J. A. (1981), "Labor Supply," in H. J. Aaron and J. A. Pechman, eds.,

How Taxes Affect Economic Behavior, The Brookings Institution,

Washington.

Hedges, J. N. and E. S. Sekscenski (1979), "Workers and Late Shifts in a

Changing Economy," Monthly Labor Review, 102, 14-22.

Ibbotson, R. G. and R. A. Sinquefeld (1982), Stocks, Bonds, Bills and

tion: The Past and the Future, Financial Analyst Research Foundation,

Charlottesville, VA.

Jorgenson,D. W. and M. A. Sullivan (1981), "Inflation and Corporate Capital

Recovery," in C. R. Hulten, ed., Depreciation, Inflation, and the

Taxation of Income from Capital, The Urban Institute, Washington. 50

Kennan,J. (1979), "The Estimation of Partial Adjustment Nodels with Rational

Expectations," Econometrica, 47, 1441-1455.

Marris, R. (1964), The Economics of Capacity Utilisation: A Report on

shift Work, Caithridge University Press, Cathridge.

______(1970),Multiple Shiftwork, National Economic Development Office

Monograph 1, HMSO, London.

Maurice, N. (1975), Shift Work: Economic Advantages and Social Cost, Inter-

national Labor Office, Geneva.

Meese, R. (1980), "Dynamic Factor Demand Schedules for Labor and Capital under

Rational Expectations," Journal of Econometrics, 14, 141-158.

Mott, P. E., F. C. Mann Q. McLoughlin, and D. P. Waick (1965), Shift Work:

The Social, Psychological, and Physical Consequences, University of

Michigan Press, Ann Arbor.

Nadiri, N. I. and S. Rosen (1969), "Interrelated Factor Demand Functions,"

American Economic Review, 59, 457-471.

Pindyck, R. S. and J. J. Roternberg (1983), "Dynamic Factor Demands under Rational

Expectations," Scandinavian Journal of Economics, 85, 223-238.

Rotemberg, J. J. (1984), "Interpreting the Statistical Failure of Some Rational

Expectations Macroeconomic Models," American Economic Review Papers and

Proceedings, 74, 188-193.

Sargent, T. J. (1978), "Estimation of Dynamic Labor Demand Schedules under

Rational Expectations," Journal of Political Economy, 86, 1009-1044.

Shapiro, H. D. (1984), "The Dynamic Demand for Capital and Labor," Cowles

Foundation Discussion Paper No. 735, New Haven, CT. Forthcoming in

Quarterly Journal of Economics.

Summers,L. H. (1981),"Taxation and Corporate Investment: A q-Theory Approach,"

BrookingsPaperson Economic Activity, No. 1, 67-127. 51

Taubman, P. and P. Gottschalk (1971) ,"TheAverage Workweek of Capital in

Manufacturing," Journal of the American Statistical Association, 66,

448—455.

Taubman, P. and M. Wilkinson (1970), "User Cost, Capacity Utilization, and

Investment Theory," International Economic Review, 11, 209-215.

Tatorn, J. A. (1980), "The 'Problem' of Real Wages and Productivity," Journal

of Political Economy, 88, 385-394.

Winston, G. C. (1974), "The Theory of Capital Utilization and Idleness,"

Journal of Economic Literature, 12, 1301-1320.

______(1982),The Timing of Economic Activities, Cambridge University

Press, Cambridge.